Counting Triangulations and other Crossing-free Structures via Onion Layers

TL;DR

This paper introduces a technique for counting crossing-free structures like triangulations and spanning cycles on point sets, with algorithms that are efficient for sets with few onion layers and outperform enumeration methods on complex instances.

Contribution

The authors develop a general method for counting crossing-free structures based on onion layers, providing algorithms with sub-exponential performance and analyzing their complexity.

Findings

Algorithms run in polynomial time for constant onion layers.

Counting triangulations is at most O*(3.1414^n), outperforming enumeration for complex instances.

The technique applies to various crossing-free structures and proves hardness for related counting problems.

Abstract

Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations of $P$, among others. In this paper we develop a general technique for computing the number of crossing-free structures of an input set $P$. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of $P$. The running time of our algorithms is upper bounded by $n^{O(k)}$, where $k$ is the number of onion layers of $P$. In particular, for $k = O(1)$ our algorithms run in polynomial time. In addition, we show that our algorithm for counting triangulations is never slower than $O^{*}(3.1414^{n})$, even when $k = \Theta(n)$. Given that there are several well-studied configurations of points with at least $\Omega(3.464^{n})$ triangulations, and some even with $\Omega(8^{n})$ triangulations, our algorithm asymptotically outperforms any enumeration algorithm for such instances. In fact, it is widely believed that any set of $n$ points must have at least $\Omega(3.464^{n})$ triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the "Restricted-Triangulation-Counting-Problem", which we prove to be $W[2]$-hard in the parameter $k$. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.